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The strict opositivity of 4th order symmetric tensor may apply to detect vacuum stability of general scalar potential. For finding analytical expressions of (strict) opositivity of 4th order symmetric tensor, we may reduce its order to 3rd order to better deal with it. So, it is provided that several analytically sufficient conditions for the copositivity of 3th order 2 dimensional (3 dimensional) symmetric tensors. Subsequently, applying these conclusions to 4th order tensors, the analytically sufficient conditions of copositivity are proved for 4th order 2 dimensional and 3 dimensional symmetric tensors. Finally, we apply these results to present analytical vacuum stability conditions for vacuum stability for $\mathbb{Z}_3$ scalar dark matter., Comment: 21 pages

In this paper, we introduce the concept of an $m$-order $n$-dimensional generalized Hilbert tensor $\mathcal{H}_{n}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}})$, $$ \mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots i_{m}-m+a},\ a\in \mathbb{R}\setminus\mathbb{Z}^-;\ i_{1},i_{2},\cdots,i_{m}=1,2,\cdots,n, $$ and show that its $H$-spectral radius and its $Z$-spectral radius are smaller than or equal to $M(a)n^{m-1}$ and $M(a)n^{\frac{m}{2}}$, respectively, here $M(a)$ is a constant only dependent on $a$. Moreover, both infinite and finite dimensional generalized Hilbert tensors are positive definite for $a\geq1$. For an $m$-order infinite dimensional generalized Hilbert tensor $\mathcal{H}_{\infty}$ with $a>0$, we prove that $\mathcal{H}_{\infty}$ defines a bounded and positively $(m-1)$-homogeneous operator from $l^{1}$ into $l^{p}\ (1, Comment: 15 pages

High order tensor arises more and more often in signal processing, data analysis, higher-order statistics, as well as imaging sciences. In this paper, an adaptive gradient (AG) method is presented for generalized tensor eigenpairs. Global convergence and linear convergence rate are established under some suitable conditions. Numerical results are reported to illustrate the efficiency of the proposed method. Comparing with the GEAP method, an adaptive shifted power method proposed by Kolda and Mayo (SIAM J Matrix Anal Appl 35:1563---1581, 2014) the AG method is much faster and could reach the largest eigenpair with a higher probability.

The Fredholm alternative type results are proved for eigenvalues ($E$-eigenvalues, $H$-eigenvalues, $Z$-eigenvalues) of a higher order tensor $\mathcal{A}$. For the positively homogeneous operators...

The main objectives of this paper are to employ a new proof technique to prove the strong convergence of { x n } and { y n } , defined respectively by x n + 1 = α n u + β n x n + ( 1 − α n − β n ) J r n A x n and y n + 1 = β n y n + ( 1 − β n ) J r n A ( α n u + ( 1 − α n ) y n ) , to some zero of accretive operator A in a uniformly convex Banach space E with a uniformly Gâteaux differentiable norm (or with a weakly continuous duality mapping J φ ) whenever { α n } and { β n } are sequences in ( 0 , 1 ) and { r n } ⊂ ( 0 , + ∞ ) satisfying the conditions: lim n → ∞ α n = 0 , ∑ n = 1 + ∞ α n = + ∞ , lim sup n → ∞ β n 1 , lim inf n → ∞ r n > 0 .

Strong convergence of two iterative schemes is proved to approach some zero of multivalued accretive operators in a Banach space. The first one is a regularization method for Rockafellar's proximal point algorithm of the resolvent and the second one is a kind of Halpern type iteration process of the resolvent. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Let f be a fixed strongly pseudocontractive mapping and T be a continuous pseudocontractive mapping with F ( T ) ≠ 0 . The sequence { z m } is iteratively defined as follows: z m = t m f ( z m ) + ( 1 − t m ) 1 m + 1 ∑ j = 0 m T j z m , m ≥ 0 , where { t m } ⊂ ( 0 , 1 ) satisfies the condition lim m → ∞ t m = 0 . We prove that { z m } converges strongly to the unique solution p to some variational inequality in F ( T ) . Our results develop and complement the corresponding ones by Matsushita–Daishi Kuroiwa [J. Math. Anal. Appl. 294 (2004) 206–214] and Shioji–Takahashi [Arch. Math., 72 (1999) 354–359] and Moore–Nnoli [J. Math. Anal. Appl. 260 (2001) 269–278] and Su–Li [Appl. Math. Comput. 181 (2006) 332–341] and Song–Chen [Appl. Math. Comput. 186 (2007) 1120–1128] and Wangkeeree [Appl. Math. Comput. 201 (2008) 239–249].

Many problems encountered in applied mathematics can be recast as the problem of selecting a particular common fixed point of a family of nonexpansive mappings in a Banach space. In this paper, under the hypotheses that E is a reflexive and strictly convex Banach spaces with a uniformly Gâeaux differentiable norm, the strong convergence of a iteration algorithm is proved for finding a common fixed point of a finite family of nonexpansive self-mappings defined on a closed convex subset K of E .

Fixed point and coincidence results are presented for single-valued hybrid generalized φ-weak contractions T,S defined on complete metric spaces.

The aim of this work is to establish the strong convergence of explicit viscosity-like methods for a nonexpansive nonself-mapping without boundary conditions defined in a Banach space. This gets rid of the restriction and dependence on the implicit anchor-like continuous path xt in the existing literature.

Let K be a nonempty closed convex subset of a reflexive and strictly convex Banach space E with a uniformly Gâteaux differentiable norm, and F = { T ( t ) : t > 0 } a nonexpansive self-mappings semigroup of K, and f : K → K a fixed contractive mapping. The strongly convergent theorems of the following implicit and explicit viscosity iterative schemes { x n } are proved. x n = α n f ( x n ) + ( 1 − α n ) T ( t n ) x n , x n + 1 = α n f ( x n ) + ( 1 − α n ) T ( t n ) x n . And the cluster point of { x n } is the unique solution to some co-variational inequality.

Let K be a nonempty closed convex subset of a Banach space E , T : K → K a continuous pseudo-contractive mapping. Suppose that { α n } is a real sequence in [ 0 , 1 ] satisfying appropriate conditions; then for arbitrary x 0 ∈ K , the Mann type implicit iteration process { x n } given by x n = α n x n − 1 + ( 1 − α n ) T x n , n ≥ 0 , strongly and weakly converges to a fixed point of T , respectively.

We proved several strong convergence results by using the conception of a uniformly asymptotically regular sequence {T n } of nonexpansive mappings in a reflexive Banach space which admits a weakly continuous duality mapping J ϕ(l p (1

Many problems encountered in applied mathematics can be recast as the problem of selecting a particular common fixed point of a countable family of non-expansive mappings in Banach space. In the paper, under the hypotheses that E is a reflexive Banach space which has a weakly continuous duality mapping Jφ with gauge function φ, several viscosity approximation methods are investigated for finding a common fixed point of non-expansive mapping sequences.

In a real reflexive and strictly convex Banach space E with a uniformly Gâteaux differentiable norm, we introduced the viscosity iterative process {xt} given by H.-K. Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291] to continuous pseudocontractive self-mapping, and proved that the {xt} strongly converges x∗∈F(T) as t→0 and x∗ is a unique solution in F(T) to the following variational inequality: 〈f(x∗)−x∗,j(z−x∗)〉≤0for all z∈F(T). We also proposed the modified implicit iteration process {xn} for continuous pseudocontractive self-mapping, and show that {xn} strongly converges x∗∈F(T). The results presented extended and improved the corresponding results of H.-K. Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291].

In this paper, we defined viscosity iterative sequence {zm} and {xn} of the Cesaro means for non-expansive mappings T, and proved that {zm} and {xn} converge strongly to some p ∈ F(T), respectively, where p is a unique solution in F(T) to the following variational inequality: 〈 ( f - I ) p , j ( u - p ) 〉 ⩽ 0 for all u ∈ F ( T ) . Our results developed and complemented the corresponding ones by Shin-ya Matsushita and Daishi Kuroiwa [Strong convergence of averaging iterations of nonexpansive nonself-mappings, J. Math. Anal. Appl. 294 (2004) 206–214] and H.K. Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291] and Yongfu Su and Suhong Li [Strong convergence theorems on two iterative method for non-expansive mappings, Appl. Math. Comput. Available from: (accessed 28.02.06)].

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E ∗ , and K be a nonempty closed convex subset of E . Suppose that { T n } ( n = 1 , 2 , … ) is a uniformly asymptotically regular sequence of nonexpansive mappings from K into itself such that F ≔ ⋂ n = 1 ∞ F ( T n ) ≠ 0 . For arbitrary initial value x 0 ∈ K and fixed contractive mapping f : K → K , define iteratively a sequence { x n } as follows: x n + 1 = λ n + 1 f ( x n ) + ( 1 − λ n + 1 ) T n + 1 x n , n ≥ 0 , where { λ n } ⊂ ( 0 , 1 ) satisfies lim n → ∞ λ n = 0 and ∑ n = 1 ∞ λ n = ∞ . We prove that { x n } converges strongly to p ∈ F , as n → ∞ , where p is the unique solution in F to the following variational inequality: 〈 ( I − f ) p , j ( p − u ) 〉 ≤ 0 for all u ∈ F ( T ) . Our results extend and improve the corresponding ones given by O’Hara et al. [J.G. O’Hara, P. Pillay, H.-K. Xu, Iterative approaches to finding nearest common fixed point of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417–1426], J.S. Jung [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509–520], H.K. Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291] and O’Hara et al. [J.G. O’Hara, P. Pillay, H.-K. Xu, Iterative approaches to convex feasibility problem in Banach space, Nonlinear Anal. Available online 20 October 2005. doi:10.1016/j.na.2005.07.36 ].

In this paper, we obtain some results on Banach operator pair better than those given by J. Chen and Z. Li [Common Fixed-points for Banach Operator Pairs in Best Approximation, J.Math. Anal. Appl. (2007), doi:10.1016/j.jmaa.2007.01.064]. Mathematics Subject Classification: 41A50, 47H10, 54H25

The tensor complementarity problem $(\q, \mathcal{A})$ is to $$\mbox{ find } \x \in \mathbb{R}^n\mbox{ such that }\x \geq \0, \q + \mathcal{A}\x^{m-1} \geq \0, \mbox{ and }\x^\top (\q + \mathcal{A}\x^{m-1}) = 0.$$ We prove that a real tensor $\mathcal{A}$ is a (strictly) semi-positive tensor if and only if the tensor complementarity problem $(\q, \mathcal{A})$ has a unique solution for $\q>\0$ ($\q\geq\0$), and a symmetric real tensor is a (strictly) semi-positive tensor if and only if it is (strictly) copositive. That is, for a strictly copositive symmetric tensor $\mathcal{A}$, the tensor complementarity problem $(\q, \mathcal{A})$ has a solution for all $\q \in \mathbb{R}^n$.

Let E a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E ∗ , and K be a closed convex subset of E which is also a sunny nonexpansive retract of E, and T : K → E be nonexpansive mappings satisfying the weakly inward condition and F ( T ) ≠ ∅ , and f : K → K be a fixed contractive mapping. The implicit iterative sequence { x t } is defined by for t ∈ ( 0 , 1 ) x t = P ( t f ( x t ) + ( 1 − t ) T x t ) . The explicit iterative sequence { x n } is given by x n + 1 = P ( α n f ( x n ) + ( 1 − α n ) T x n ) , where α n ∈ ( 0 , 1 ) and P is sunny nonexpansive retraction of E onto K. We prove that { x t } strongly converges to a fixed point of T as t → 0 , and { x n } strongly converges to a fixed point of T as α n satisfying appropriate conditions. The results presented extend and improve the corresponding results of [H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291].

Let K be a closed convex subset of a q-uniformly smooth separable Banach space, T: K → K a strictly pseudocontractive mapping, and f: K → K an L-Lispschitzian strongly pseudocontractive mapping. For any t ∈ (0, 1), let xt be the unique fixed point of t f + (1 - t)T. We prove that if T has a fixed point, then {xt} converges to a fixed point of T as t approaches to 0.

It is easily checkable if a given tensor is a B tensor, or a B$_0$ tensor or not. In this paper, we show that a symmetric B tensor can always be decomposed to the sum of a strictly diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors, and a symmetric B$_0$ tensor can always be decomposed to the sum of a diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors. When the order is even, this implies that the corresponding B tensor is positive definite, and the corresponding B$_0$ tensor is positive semi-definite. This gives a checkable sufficient condition for positive definite and semi-definite tensors. This approach is different from the approach in the literature for proving a symmetric B matrix is positive definite, as that matrix approach cannot be extended to the tensor case.

In this paper, we extend some classes of structured matrices to higher order tensors. We discuss their relationships with positive semi-definite tensors and some other structured tensors. We show that every principal sub-tensor of such a structured tensor is still a structured tensor in the same class, with a lower dimension. The potential links of such structured tensors with optimization, nonlinear equations, nonlinear complementarity problems, variational inequalities and the nonnegative tensor theory are also discussed., arXiv admin note: text overlap with arXiv:1405.1288 by other authors

For an m-order n-dimensional Hilbert tensor (hypermatrix) H n = ( H i 1 i 2 ⋯ i m ) , H i 1 i 2 ⋯ i m = 1 i 1 + i 2 + ⋯ + i m − m + 1 , i 1 , … , i m = 1 , 2 , … , n its spectral radius is not larger than n m − 1 sin π n , and an upper bound of its E-spectral radius is n m 2 sin π n . Moreover, its spectral radius is strictly increasing and its E-spectral radius is nondecreasing with respect to the dimension n. When the order is even, both infinite and finite dimensional Hilbert tensors are positive definite. We also show that the m-order infinite dimensional Hilbert tensor (hypermatrix) H ∞ = ( H i 1 i 2 ⋯ i m ) defines a bounded and positively ( m − 1 ) -homogeneous operator from l 1 into l p ( 1 p ∞ ), and the norm of corresponding positively homogeneous operator is smaller than or equal to π 6 .

We discuss the following viscosity approximations with the weak contraction for a non-expansive mapping sequence , , . We prove that Browder's and Halpern's type convergence theorems imply Moudafi's viscosity approximations with the weak contraction, and give the estimate of convergence rate between Halpern's type iteration and Mouda's viscosity approximations with the weak contraction.

In this paper, a necessary and sufficient conditions for the strong convergence to a common fixed point of a finite family of continuous pseudocontractive mappings are proved in an arbitrary real Banach space using an implicit iteration scheme recently introduced by Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Fuct. Anal. Optim. 22 (2001) 767–773] in condition α n ∈ ( 0 , 1 ] , and also strong and weak convergence theorem of a finite family of strictly pseudocontractive mappings of Browder–Petryshyn type is obtained. The results presented extend and improve the corresponding results of M.O. Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73–81].

An iterative algorithm is proposed for finding a fixed point of a nonexpansive self-mapping of a closed convex subset of a Banach space with a uniformly Gâteaux differentiable norm, and an estimate of the convergence speed also is given.